| Conjecture | Field | Comments | Eponym(s) |
|---|
| 1/3–2/3 conjecture | order theory | n/a |
| abc conjecture | number theory | ⇔Granville–Langevin conjecture, Vojta's conjecture in dimension 1
⇒Erdős–Woods conjecture, Fermat–Catalan conjecture
Formulated by David Masser and Joseph Oesterlé.[1]
Proof claimed in 2012 by Shinichi Mochizuki | n/a |
| Agoh–Giuga conjecture | number theory | Takashi Agoh and Giuseppe Giuga |
| Agrawal's conjecture | number theory | Manindra Agrawal |
| Andrews–Curtis conjecture | combinatorial group theory | James J. Andrews and Morton L. Curtis |
| Andrica's conjecture | number theory | Dorin Andrica |
| Artin conjecture (L-functions) | number theory | Emil Artin |
| Artin's conjecture on primitive roots | number theory | Emil Artin |
| Bateman–Horn conjecture | number theory | Paul T. Bateman and Roger Horn |
| Baum–Connes conjecture | operator K-theory |
| Beal's conjecture | number theory |
| Beilinson conjecture | number theory |
| Berry–Tabor conjecture | geodesic flow |
| Birch and Swinnerton-Dyer conjecture | number theory |
| Birch–Tate conjecture | number theory |
| Birkhoff conjecture | integrable systems |
| Bloch–Beilinson conjectures | number theory |
| Bloch–Kato conjecture | algebraic K-theory |
| Bombieri–Lang conjecture | diophantine geometry | Enrico Bombieri and Serge Lang |
| Borel conjecture | geometric topology | Armand Borel |
| Bost conjecture | geometric topology |
| Brennan conjecture | complex analysis |
| Brocard's conjecture | number theory |
| Brumer–Stark conjecture | number theory |
| Bunyakovsky conjecture | number theory |
| Carathéodory conjecture | differential geometry |
| Carmichael totient conjecture | number theory |
| Casas-Alvero conjecture | polynomials |
| Catalan–Dickson conjecture on aliquot sequences | number theory |
| Catalan's Mersenne conjecture | number theory |
| Cherlin–Zilber conjecture | group theory |
| Chowla conjecture | Möbius function | ⇒Sarnak conjecture[2] | Sarvadaman Chowla |
| Collatz conjecture | number theory |
| Cramér's conjecture | number theory |
| Conway's thrackle conjecture | graph theory | John Horton Conway |
| Deligne conjecture | monodromy | Pierre Deligne |
| Dittert conjecture | combinatorics |
| Eilenberg−Ganea conjecture | algebraic topology |
| Elliott–Halberstam conjecture | number theory | Peter D. T. A. Elliott and Heini Halberstam |
| Erdős–Faber–Lovász conjecture | graph theory |
| Erdős–Gyárfás conjecture | graph theory |
| Erdős–Straus conjecture | number theory |
| Farrell–Jones conjecture | geometric topology |
| Filling area conjecture | differential geometry |
| Firoozbakht's conjecture | number theory |
| Fortune's conjecture | number theory |
| Frankl conjecture | combinatorics |
| Gauss circle problem | number theory |
| Gilbreath conjecture | number theory |
| Goldbach's conjecture | number theory | ⇒The ternary Goldbach conjecture, which was the original formulation.[3] | Christian Goldbach |
| Gold partition conjecture[4] | order theory |
| Goormaghtigh conjecture | number theory |
| Green's conjecture | algebraic curves |
| Grimm's conjecture | number theory |
| Grothendieck–Katz p-curvature conjecture | differential equations | Alexander Grothendieck and Nicholas Katz |
| Hadamard conjecture | combinatorics |
| Hedetniemi's conjecture | graph theory |
| Herzog–Schönheim conjecture | group theory |
| Hilbert–Smith conjecture | geometric topology |
| Hodge conjecture | algebraic geometry |
| Homological conjectures in commutative algebra | commutative algebra |
| Hopf conjectures | geometry | Heinz Hopf |
| Invariant subspace problem | functional analysis | n/a |
| Jacobian conjecture | polynomials |
| Jacobson's conjecture | ring theory | Nathan Jacobson |
| Kakeya conjectures | geometry |
| Kaplansky conjectures | ring theory | Irving Kaplansky |
| Keating–Snaith conjecture | number theory |
| Köthe conjecture | ring theory |
| Kung–Traub conjecture | iterative methods |
| Legendre's conjecture | number theory |
| Lemoine's conjecture | number theory |
| Lenstra–Pomerance–Wagstaff conjecture | number theory |
| Leopoldt's conjecture | number theory |
| List coloring conjecture | graph theory | n/a |
| Littlewood conjecture | diophantine approximation | John Edensor Littlewood |
| Lovász conjecture | graph theory |
| MNOP conjecture | algebraic geometry | n/a |
| Manin conjecture | diophantine geometry | Yuri Manin |
| Marshall Hall's conjecture | number theory | Marshall Hall, Jr. |
| Mazur's conjectures | diophantine geometry |
| Montgomery's pair correlation conjecture | number theory | Hugh Montgomery |
| n conjecture | number theory | n/a |
| New Mersenne conjecture | number theory |
| Novikov conjecture | algebraic topology | Sergei Novikov |
| Oppermann's conjecture | number theory |
| Petersen coloring conjecture | graph theory |
| Pierce–Birkhoff conjecture | real algebraic geometry |
| Pillai's conjecture | number theory |
| De Polignac's conjecture | number theory |
| quantum unique ergodicity conjecture | dynamical systems | 2004, Elon Lindenstrauss, for arithmetic hyperbolic surfaces,[5] 2008, Kannan Soundararajan & Roman Holowinsky, for holomorphic forms of increasing weight for Hecke eigenforms on noncompact arithmetic surfaces[6] | n/a |
| Reconstruction conjecture | graph theory | n/a |
| Riemann hypothesis | number theory | ⇐Generalized Riemann hypothesis⇐Grand Riemann hypothesis
⇔De Bruijn–Newman constant=0
⇒density hypothesis, Lindelöf hypothesis
See Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems). | Bernhard Riemann |
| Ringel–Kotzig conjecture | graph theory |
| Rudin's conjecture | additive combinatorics | Walter Rudin |
| Sarnak conjecture | topological entropy | Peter Sarnak |
| Sato–Tate conjecture | number theory |
| Schanuel's conjecture | number theory |
| Schinzel's hypothesis H | number theory | Andrzej Schinzel |
| Scholz conjecture | addition chains |
| Second Hardy–Littlewood conjecture | number theory | G. H. Hardy and J. E. Littlewood |
| Selfridge's conjecture | number theory |
| Sendov's conjecture | complex polynomials |
| Serre's multiplicity conjectures | commutative algebra | Jean-Pierre Serre |
| Singmaster's conjecture | binomial coefficients | David Singmaster |
| Standard conjectures on algebraic cycles | algebraic geometry | n/a |
| Tate conjecture | algebraic geometry | John Tate |
| Toeplitz' conjecture | Jordan curves | Otto Toeplitz |
| Twin prime conjecture | number theory | n/a |
| Ulam's packing conjecture | packing | Stanislas Ulam |
| Unicity conjecture for Markov numbers | number theory | n/a |
| Uniformity conjecture | diophantine geometry | n/a |
| Unique games conjecture | number theory | n/a |
| Vandiver's conjecture | number theory |
| Vizing's conjecture | graph theory |
| Waring's conjecture | number theory | Edward Waring |
| Weight monodromy conjecture | algebraic geometry | n/a |
| Weinstein conjecture | periodic orbits |
| Whitehead conjecture | algebraic topology | J. H. C. Whitehead |
| Zauner's conjecture | operator theory |
| Zhou conjecture | number theory |
The conjecture terminology may persist: theorems may not be their "official" names.
| Priority date | Proved by | Former name | Field | Comments |
|---|
| 1971 | Daniel Quillen | Adams conjecture | algebraic topology | On the J-homomorphism, proposed 1963 by Frank Adams |
| 1973 | Pierre Deligne | Weil conjectures | algebraic geometry | ⇒Ramanujan–Petersson conjecture
Proposed by André Weil. Deligne's theorems completed around 15 years of work on the general case. |
| 1983 | Gerd Faltings | Mordell conjecture | number theory | ⇐Faltings's theorem, the Shafarevich conjecture on finiteness of isomorphism classes of abelian varieties. The reduction step was by Alexey Parshin. |
| 1989 | V. I. Chernousov | Weil's conjecture on Tamagawa numbers | algebraic groups | The problem, based on Siegel's theory for quadratic forms, submitted to a long series of case analysis steps. |
| 1994 | David Harbater and Michel Raynaud | Abhyankar's conjecture | algebraic geometry |
| 1994 | Andrew Wiles | Fermat's Last Theorem | number theory | ⇔The modularity theorem for semistable elliptic curves.
Proof completed with Richard Taylor. |
| 1995 | Doron Zeilberger[7] | Alternating sign matrix conjecture, | enumerative combinatorics |
| 2001 | Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor | Taniyama–Shimura conjecture | elliptic curves | Now the modularity theorem for elliptic curves. Once known as the "Weil conjecture". |
| 2013 | Zhang Yitang | bounded gap conjecture | number theory | The sequence of gaps between consecutive prime numbers has a finite lim inf. See Polymath Project#Polymath8 for quantitative results. |
1. ^{{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2002 |publisher=CRC Press |isbn=9781420035223 |page=13 |url=https://books.google.co.uk/books?id=D_XKBQAAQBAJ&pg=PA13 |language=en}}
2. ^{{cite web |last1=Tao |first1=Terence |title=The Chowla conjecture and the Sarnak conjecture |url=https://terrytao.wordpress.com/2012/10/14/the-chowla-conjecture-and-the-sarnak-conjecture/ |website=What's new |language=en |date=15 October 2012}}
3. ^{{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2002 |publisher=CRC Press |isbn=9781420035223 |page=1203 |url=https://books.google.co.uk/books?id=D_XKBQAAQBAJ&pg=PA1203 |language=en}}
4. ^M. Peczarski, The gold partition conjecture, it Order 23(2006): 89–95.
5. ^{{cite web|url=http://www.math.kth.se/4ecm/prizes.ecm.html|title=EMS Prizes|website=www.math.kth.se}}
6. ^{{cite web |url=http://matematikkforeningen.no/INFOMAT/08/0810.pdf |title=Archived copy |accessdate=2008-12-12 |deadurl=yes |archiveurl=https://web.archive.org/web/20110724181506/http://matematikkforeningen.no/INFOMAT/08/0810.pdf |archivedate=2011-07-24 |df= }}
7. ^{{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2002 |publisher=CRC Press |isbn=9781420035223 |page=65 |url=https://books.google.co.uk/books?id=D_XKBQAAQBAJ&pg=PA65 |language=en}}
8. ^{{cite book |last1=Weisstein |first1=Eric W. |title=CRC Concise Encyclopedia of Mathematics |date=2002 |publisher=CRC Press |isbn=9781420035223 |page=218 |url=https://books.google.co.uk/books?id=D_XKBQAAQBAJ&pg=PA218 |language=en}}
9. ^Daniel Frohardt and Kay Magaard, Composition Factors of Monodromy Groups, Annals of Mathematics Second Series, Vol. 154, No. 2 (Sep., 2001), pp. 327–345. Published by: Mathematics Department, Princeton University DOI: 10.2307/3062099 {{jstor|3062099}}
10. ^{{SpringerEOM|title=Schoenflies conjecture|id=Schoenflies_conjecture}}